Theory and application of specular path perturbation

Abstract

In this paper we apply perturbation methods to the problem of computing specular reflections in curved surfaces. The key idea is to generate families of closely related optical paths by expanding a given path into a high-dimensional Taylor series. Our path perturbation method is based on closed-form expressions for linear and higher-order approximations of ray paths, which are derived using Fermat's Variation Principle and the Implicit Function Theorem (IFT). The perturbation formula presented here holds for general multiple-bounce reflection paths and provides a mathematical foundation for exploiting path coherence in ray tracing acceleration techniques and incremental rendering. To illustrate its use, we describe an algorithm for fast approximation of specular reflections on curved surfaces; the resulting images are highly accurate and nearly indistinguishable from ray traced images. Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/Image Generation -Display algorithms; I.3.6 [Computer Graphics]: Methodology and Techniques - Interaction techniques; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism - Raytracing; G.1.1 [Numerical Analysis]: Interpolation - Interpolation formulas; G.1.2 [Numerical Analysis]: Approximation - Elementary function approximation.